# Properties

 Label 6027.2.a.y Level 6027 Weight 2 Character orbit 6027.a Self dual Yes Analytic conductor 48.126 Analytic rank 0 Dimension 7 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$6027 = 3 \cdot 7^{2} \cdot 41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6027.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$48.1258372982$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} - q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} -\beta_{5} q^{5} + ( -1 + \beta_{1} ) q^{6} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} - q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} -\beta_{5} q^{5} + ( -1 + \beta_{1} ) q^{6} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} + q^{9} + ( 1 - \beta_{3} - \beta_{5} ) q^{10} + ( 2 + \beta_{4} - \beta_{5} ) q^{11} + ( -2 + \beta_{1} - \beta_{2} ) q^{12} + ( 1 - \beta_{6} ) q^{13} + \beta_{5} q^{15} + ( 3 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{16} + ( -2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{17} + ( 1 - \beta_{1} ) q^{18} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{19} + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{20} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{22} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{23} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{24} + ( -1 - 2 \beta_{2} + \beta_{3} - \beta_{6} ) q^{25} + ( 3 - \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{26} - q^{27} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{29} + ( -1 + \beta_{3} + \beta_{5} ) q^{30} + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{31} + ( 4 + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{32} + ( -2 - \beta_{4} + \beta_{5} ) q^{33} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{6} ) q^{34} + ( 2 - \beta_{1} + \beta_{2} ) q^{36} + ( 2 + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{37} + ( -2 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{38} + ( -1 + \beta_{6} ) q^{39} + ( 4 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{40} - q^{41} + ( \beta_{3} + \beta_{4} + \beta_{6} ) q^{43} + ( 5 - \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{44} -\beta_{5} q^{45} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{46} + ( -2 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{47} + ( -3 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{48} + ( 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{50} + ( 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{51} + ( 7 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{6} ) q^{52} + ( 3 - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{53} + ( -1 + \beta_{1} ) q^{54} + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{55} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{6} ) q^{57} + ( -1 + \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{58} + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{59} + ( -3 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{60} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{61} + ( 2 + \beta_{3} - \beta_{5} + \beta_{6} ) q^{62} + ( 1 - \beta_{3} - \beta_{4} - 3 \beta_{6} ) q^{64} + ( 4 - \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{65} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{66} + ( -1 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{67} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{68} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{69} + ( 1 + \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{71} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{72} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{73} + ( 2 - 3 \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{74} + ( 1 + 2 \beta_{2} - \beta_{3} + \beta_{6} ) q^{75} + ( 6 + \beta_{2} + \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{76} + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{78} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{79} + ( 7 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{80} + q^{81} + ( -1 + \beta_{1} ) q^{82} + ( 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{83} + ( -2 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{85} + ( -5 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{86} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{87} + ( 7 - 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{6} ) q^{88} + ( -7 + \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{89} + ( 1 - \beta_{3} - \beta_{5} ) q^{90} + ( -4 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + 5 \beta_{6} ) q^{92} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{93} + ( -7 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{94} + ( 3 \beta_{2} - \beta_{3} + 4 \beta_{5} ) q^{95} + ( -4 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{96} + ( 6 - \beta_{1} - 4 \beta_{3} + \beta_{4} + 2 \beta_{6} ) q^{97} + ( 2 + \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + 4q^{2} - 7q^{3} + 8q^{4} - q^{5} - 4q^{6} + 12q^{8} + 7q^{9} + O(q^{10})$$ $$7q + 4q^{2} - 7q^{3} + 8q^{4} - q^{5} - 4q^{6} + 12q^{8} + 7q^{9} + 3q^{10} + 11q^{11} - 8q^{12} + 7q^{13} + q^{15} + 6q^{16} - 11q^{17} + 4q^{18} - 4q^{19} + 7q^{20} + 6q^{22} + 7q^{23} - 12q^{24} + 2q^{25} + 13q^{26} - 7q^{27} + 4q^{29} - 3q^{30} + 7q^{31} + 18q^{32} - 11q^{33} + 20q^{34} + 8q^{36} - 4q^{38} - 7q^{39} + 9q^{40} - 7q^{41} + q^{43} + 18q^{44} - q^{45} - 17q^{46} - 14q^{47} - 6q^{48} + 19q^{50} + 11q^{51} + 27q^{52} + 23q^{53} - 4q^{54} + 30q^{55} + 4q^{57} - 3q^{58} - 8q^{59} - 7q^{60} + 3q^{61} + 16q^{62} + 6q^{64} + 15q^{65} - 6q^{66} + 3q^{67} - 7q^{69} + 7q^{71} + 12q^{72} + 11q^{73} - 13q^{74} - 2q^{75} + 40q^{76} - 13q^{78} - q^{79} + 43q^{80} + 7q^{81} - 4q^{82} - 10q^{85} - 12q^{86} - 4q^{87} + 10q^{88} - 32q^{89} + 3q^{90} - 19q^{92} - 7q^{93} - 21q^{94} - 8q^{95} - 18q^{96} + 25q^{97} + 11q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3 x^{6} - 6 x^{5} + 16 x^{4} + 14 x^{3} - 20 x^{2} - 10 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 3 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} - 3 \nu^{5} - 4 \nu^{4} + 12 \nu^{3} + 4 \nu^{2} - 8 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} - 3 \nu^{5} - 6 \nu^{4} + 16 \nu^{3} + 12 \nu^{2} - 16 \nu - 4$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} + 5 \nu^{5} - 2 \nu^{4} - 18 \nu^{3} + 14 \nu^{2} + 14 \nu - 6$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$-\beta_{5} + \beta_{4} + 2 \beta_{3} + 8 \beta_{2} + 10 \beta_{1} + 16$$ $$\nu^{5}$$ $$=$$ $$\beta_{6} - 3 \beta_{5} + 4 \beta_{4} + 9 \beta_{3} + 21 \beta_{2} + 33 \beta_{1} + 33$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{6} - 13 \beta_{5} + 18 \beta_{4} + 23 \beta_{3} + 67 \beta_{2} + 83 \beta_{1} + 115$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.95889 2.35311 1.19009 0.281557 −0.762978 −1.32604 −1.69463
−1.95889 −1.00000 1.83724 0.513172 1.95889 0 0.318823 1.00000 −1.00525
1.2 −1.35311 −1.00000 −0.169102 −1.31916 1.35311 0 2.93503 1.00000 1.78497
1.3 −0.190092 −1.00000 −1.96387 −2.28332 0.190092 0 0.753499 1.00000 0.434041
1.4 0.718443 −1.00000 −1.48384 3.61951 −0.718443 0 −2.50294 1.00000 2.60041
1.5 1.76298 −1.00000 1.10809 −3.51322 −1.76298 0 −1.57241 1.00000 −6.19372
1.6 2.32604 −1.00000 3.41045 −0.0978008 −2.32604 0 3.28076 1.00000 −0.227488
1.7 2.69463 −1.00000 5.26102 2.08082 −2.69463 0 8.78724 1.00000 5.60704
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$-1$$
$$41$$ $$1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6027))$$:

 $$T_{2}^{7} - 4 T_{2}^{6} - 3 T_{2}^{5} + 24 T_{2}^{4} - 7 T_{2}^{3} - 34 T_{2}^{2} + 15 T_{2} + 4$$ $$T_{5}^{7} + T_{5}^{6} - 18 T_{5}^{5} - 18 T_{5}^{4} + 69 T_{5}^{3} + 57 T_{5}^{2} - 36 T_{5} - 4$$ $$T_{13}^{7} - 7 T_{13}^{6} - 14 T_{13}^{5} + 114 T_{13}^{4} - 83 T_{13}^{3} - 223 T_{13}^{2} + 294 T_{13} - 72$$